Final answer:
In the context of linear transformations from R³ to R³, Rotation, Reflection, and Scaling are invertible because each of them has a corresponding reverse operation, while Projection is not typically invertible.
Step-by-step explanation:
The question involves identifying which linear transformations from R³ to R³ are invertible among the given options. A linear transformation is invertible if it has a two-sided inverse, which means applying the transformation and then applying its inverse will result in the original vector.
- A) Rotation: A rotation is an invertible transformation because each rotation has a corresponding reverse rotation that can return vectors to their original positions.
- B) Reflection: A reflection about a plane is also invertible because reflecting a point twice about the same plane will return the point to its original location.
- C) Scaling: Scaling by a non-zero scalar is invertible since the scaling can be undone by scaling by the reciprocal of the original scalar.
- D) Projection: Projection is not typically invertible because it loses information about the vector's component in the direction perpendicular to the projection plane or line. Once projected, the original vector cannot be recovered.
Therefore, the linear transformations from R³ to R³ that are invertible are Rotation, Reflection, and Scaling.